On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow

1974 ◽  
Vol 63 (3) ◽  
pp. 529-536 ◽  
Author(s):  
A. Davey ◽  
L. M. Hocking ◽  
K. Stewartson
Keyword(s):  
Pressure Gradient ◽  
Poiseuille Flow ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Governing Equations ◽  
Amplitude Function ◽  
Coupled Partial Differential Equations

The equations governing the nonlinear development of a centred three-dimensional disturbance to plane parallel flow at slightly supercritical Reynolds numbers are obtained, In contrast to the corresponding equation for two-dimensional disturbances, two slowly varying functions are needed to describe the development: the amplitude function and a function related to the secular pressure gradient produced by the disturbance. These two functions satisfy a pair of coupled partial differential equations. The equations derived in Hocking, Stewartson & Stuart (1972) are shown to be incorrect, Some of the properties of the governing equations are discussed briefly.

scholarly journals Kazantsev model in non-helical 2.5-dimensional flows

2016 ◽  
Vol 806 ◽  
pp. 627-648 ◽  
Author(s):  
K. Seshasayanan ◽  
A. Alexakis
Keyword(s):  
Turbulent Flows ◽  
Three Dimensional ◽  
Analytical Calculation ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Control Parameters ◽  
Governing Equations ◽  
Field Correlation ◽  
Good Agreement

We study the dynamo instability for a Kazantsev–Kraichnan flow with three velocity components that depend only on two dimensions $\boldsymbol{u}=(u(x,y,t),v(x,y,t),w(x,y,t))$ often referred to as 2.5-dimensional (2.5-D) flow. Within the Kazantsev–Kraichnan framework we derive the governing equations for the second-order magnetic field correlation function and examine the growth rate of the dynamo instability as a function of the control parameters of the system. In particular we investigate the dynamo behaviour for large magnetic Reynolds numbers $Rm$ and flows close to being two-dimensional and show that these two limiting procedures do not commute. The energy spectra of the unstable modes are derived analytically and lead to power-law behaviour that differs from the three-dimensional and two-dimensional cases. The results of our analytical calculation are compared with the results of numerical simulations of dynamos driven by prescribed fluctuating flows as well as freely evolving turbulent flows, showing good agreement.

A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop

2000 ◽  
Vol 411 ◽  
pp. 325-350 ◽  
Author(s):  
SAEED MORTAZAVI ◽  
GRÉTAR TRYGGVASON
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Poiseuille Flow ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Lateral Position ◽  
Physical Parameters ◽  
Computational Domain ◽  
Two Dimensional ◽  
Lateral Migration

The cross-stream migration of a deformable drop in two-dimensional Hagen–Poiseuille flow at finite Reynolds numbers is studied numerically. In the limit of a small Reynolds number (< 1), the motion of the drop depends strongly on the ratio of the viscosity of the drop fluid to the viscosity of the suspending fluid. For viscosity ratio 0.125 a drop moves toward the centre of the channel, while for ratio 1.0 it moves away from the centre until halted by wall repulsion. The rate of migration increases with the deformability of the drop. At higher Reynolds numbers (5–50), the drop either moves to an equilibrium lateral position about halfway between the centreline and the wall – according to the so-called Segre–Silberberg effect or it undergoes oscillatory motion. The steady-state position depends only weakly on the various physical parameters of the flow, but the length of the transient oscillations increases as the Reynolds number is raised, or the density of the drop is increased, or the viscosity of the drop is decreased. Once the Reynolds number is high enough, the oscillations appear to persist forever and no steady state is observed. The numerical results are in good agreement with experimental observations, especially for drops that reach a steady-state lateral position. Most of the simulations assume that the flow is two-dimensional. A few simulations of three-dimensional flows for a modest Reynolds number (Re = 10), and a small computational domain, confirm the behaviour seen in two dimensions. The equilibrium position of the three-dimensional drop is close to that predicted in the simulations of two-dimensional flow.

scholarly journals The stability of thermally stratified plane Poiseuille flow

1968 ◽  
Vol 33 (1) ◽  
pp. 21-32 ◽  
Author(s):  
K. S. Gage ◽  
W. H. Reid
Keyword(s):  
Poiseuille Flow ◽  
Richardson Number ◽  
Three Dimensional ◽  
Complete Analysis ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Plane Poiseuille Flow ◽  
Spiral Flow ◽  
Tollmien Schlichting ◽  
The Stability

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Bénard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

Instabilities of plane Poiseuille flow with a streamwise system rotation

2008 ◽  
Vol 603 ◽  
pp. 189-206 ◽  
Author(s):  
S. MASUDA ◽  
S. FUKUDA ◽  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Secondary Flow ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Mean Flow ◽  
System Rotation ◽  
Spiral Vortex

We analyse the stability of plane Poiseuille flow with a streamwise system rotation. It is found that the instability due to two-dimensional perturbations, which sets in at the well-known critical Reynolds number, Rc = 5772.2, for the non-rotating case, is delayed as the rotation is increased from zero, showing a stabilizing effect of rotation. As the rotation is increased further, however, the laminar flow becomes most unstable to perturbations which are three-dimensional. The critical Reynolds number due to three-dimensional perturbations at this higher rotation case is many orders of magnitude less than the corresponding value due to two-dimensional perturbations. We also perform a nonlinear analysis on a bifurcating three-dimensional secondary flow. The secondary flow exhibits a spiral vortex structure propagating in the streamwise direction. It is confirmed that an antisymmetric mean flow in the spanwise direction is generated in the secondary flow.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Review of Three Dimensional Dynamic Analyses of Circular Cylinders and Cylindrical Shells

1994 ◽  
Vol 47 (10) ◽  
pp. 501-516 ◽  
Author(s):  
Kostas P. Soldatos
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Circular Cylinders ◽  
Accurate Information ◽  
Two Dimensional ◽  
Governing Equations ◽  
Complicated Form ◽  
Dynamic Analyses ◽  
Cylindrical Panels ◽  
Elastic Cylinders

There is an increasing usefulness of exact three-dimensional analyses of elastic cylinders and cylindrical shells in composite materials applications. Such analyses are considered as benchmarks for the range of applicability of corresponding studies based on two-dimensional and/or finite element modeling. Moreover, they provide valuable, accurate information in cases that corresponding predictions based on that later kind of approximate modeling is not satisfactory. Due to the complicated form of the governing equations of elasticity, such three-dimensional analyses are comparatively rare in the literature. There is therefore a need for further developments in that area. A survey of the literature dealing with three-dimensional dynamic analyses of cylinders and open cylindrical panels will serve towards such developments. This paper presents such a survey within the framework of linear elasticity.

Cellular flow in a partially filled rotating drum: regular and chaotic advection

2017 ◽  
Vol 825 ◽  
pp. 631-650 ◽  
Author(s):  
Francesco Romanò ◽  
Arash Hajisharifi ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Rotating Drum ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Heteroclinic Connection ◽  
Two Dimensional Flow ◽  
Chaotic Streamlines

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

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